1. Field of the Invention
The invention relates generally to an arrangement and a method for estimating the linear and nonlinear parameters of a model describing a transducer which converts input signals (e.g., electrical, mechanical or acoustical signals) into output signals (e.g., electrical, mechanical or acoustical signals). Transducers of this kind are primarily actuators (such as loudspeakers) and sensors (such as microphones), but also electrical systems for storing, transmitting and converting signals. The model is nonlinear and describes the internal states of the transducer and the transfer behavior between input and output at small and high amplitudes. The model has free parameters which have to be identified for the particular transducer at high precision while avoiding any systematic error (bias). The identification of nonlinear systems is the basis for measurement applications, quality assessment and failure diagnostics and for controlling the transducer actively.
2. Description of the Related Art
Most of the nonlinear system identification techniques known in prior art are based on generic structures such as polynomial filters using the Volterra-Wiener-series as described by V. J. Mathews, Adaptive Polynomial Filters, IEEE SP MAGAZINE, Jul. 1991, pages 10-26. Those methods use structures with sufficient complexity and a large number of free parameters to model the real system with sufficient accuracy. This approach is not applicable to an electro-acoustical transducer as the computational load can not be processed by available digital signal processors (DSPs). However, by exploiting a priori information on physical relationships it is possible to develop special models dedicated to a particular transducer as disclosed in U.S. Pat. No. 5,438,625 and by J. Suykens, et al., “Feedback linearization of Nonlinear Distortion in Electro-dynamic Loudspeakers,” J. Audio Eng. Soc., 43, pp 690-694). Those models have a relatively low complexity and use a minimal number of states (displacement, current, voltage, etc.) and free parameters (mass, stiffness, resistance, inductance, etc.). Static and dynamic methods have been developed for measuring the parameters of those transducer-oriented models. The technique disclosed by W. Klippel, “The Mirror Filter—a New Basis for Reducing Nonlinear Distortion Reduction and Equalizing Response in Woofer Systems”, J. Audio Eng. Society 32 (1992), pp. 675-691, is based on a traditional method for measuring nonlinear distortion. The excitation signal is a two-tone signal generating sparse distortion components which can be identified as harmonic, summed-tone or difference tone components of a certain order. This method is time consuming and can not be extended to a multi-tone stimulus because the distortion components interfere if the number of fundamental tones is high. In order to estimate the nonlinear parameters with an audio-like signal (e.g., music), adaptive methods have been disclosed in DE 4332804A1 or W. Klippel, “Adaptive Nonlinear Control of Loudspeaker Systems,” J Audio Eng. Society 46, pp. 939-954 (1998).
Patents DE 4334040, WO 97/25833, US 2003/0118193, U.S. Pat. Nos. 6,269,318 and 5,523,715 disclose control systems based on the measurement of current and voltage at the loudspeaker terminals while dispensing with an additional acoustical or mechanical sensor.
Other identification methods, such as those disclosed in U.S. Pat. Nos. 4,196,418, 4,862,160, 5,539,482, EP1466289, U.S. Pat. Nos. 5,268,834, 5,266,875, 4,291,277, EP1423664, U.S. Pat. No. 6,611,823, WO 02/02974, WO 02/095650, provide only optimal estimates for the model parameters if the model describes the behavior of the transducer completely. However, there are always differences between the theoretical model and the real transducer which causes significant errors in the estimated nonlinear parameters (bias). This shall be described in the following section in greater detail:
The output signal y(t) of the transducer:y(t)=ynlin(t)+ylin(t)  (1)consists of a nonlinear signal part:ynlin(t)=PsnGn(t)  (2)and a linear signal part:ylin(t)=PslGl(t)+er(t).  (3)
The linear signal part ylin(t) comprises a scalar product PslGl(t) of a linear parameter vector Psl, a gradient vector Gl(t) and a residual signal er(t) due to measurement noise and imperfections of the model.
The nonlinear signal part ynlin(t) can be interpreted as nonlinear distortion and can be described as a scalar product of the parameter vector:Psn=[ps,1 ps,2 . . . ps,N]  (4)and the gradient vector:GnT(t)=[g1(t)g2(t) . . . gN(t)]  (5)which may contain, for example:GnT(t)=[i(t)x(t)2i(t)x(t)4i(t)x6]  (6)products of input signal x(t) and the input current:i(t)=hi(t)*x(t).  (7)The model generates an output signal:y′(t)=PnGn(t)+PlGl(t),  (8)which comprises scalar products of the nonlinear parameter vector:Pn=[Pn,1 Pn,2 . . . Pn,N]  (9)and the linear parameter vector:Pl=[Pl,1 Pl,2 . . . Pl,L]  (10)with the corresponding linear and nonlinear gradient vector Gn(t) and Gl(t), respectively. It is the target of the optimal system identification that the parameters of the model coincide with the true parameters of the transducer (Pn→Ps,n, Pl→Ps,l).
A suitable criterion for the agreement between model and reality is the error time signal:e(t)=y(t)−y′(t),  (11)which can be represented as a sum:e(t)=en(t)+el(t)+er(t)  (12)comprising a nonlinear error part:en(t)=ΔPnGn=(Psn−Pn)Gn,  (13)a linear error part:el(t)=ΔPlGl(Psl−Pl)Gl  (14)and the residual signal er(t).
System identification techniques known in the prior art determine the linear and nonlinear parameters of the model by minimizing the total error e(t) in a cost function:C=E{e(t)2}→Minimum.  (15)
The linear parameters Pl are estimated by inserting Eqs. (1) and (8) into Eq. (11), multiplying with the transposed gradient vector GlT(t) and calculating the expectation value
                                          E            ⁢                          {              f              }                                =                                    lim                              T                →                ∞                                      ⁢                                          (                                                      1                    T                                    ⁢                                                            ∫                      0                      T                                        ⁢                                          f                      ⁡                                              (                        t                        )                                                                                            )                            .                                      ⁢                                                      (        16        )            
Considering that the residual error er(t) is not correlated with the linear gradient signals in Gl(t), this results in the Wiener-Hopf-equation:PlE{Gl(t)GlT(t)}=E{ylin(t)GlT(t)}−E{en(t)GlT(t)}PlSGGl=SyGl+SrGl  (17)which can be solved directly by multiplying this equation with the inverted matrix SGGl:Pl=(SyGl+SrGl)SGGl−1Pl=Psl+ΔPl  (18)or determined iteratively by using the LMS algorithm:
                                                                                          P                  l                  T                                ⁡                                  (                  t                  )                                            =                            ⁢                                                                    P                    l                    T                                    ⁡                                      (                                          t                      -                      1                                        )                                                  +                                  μ                  ⁢                                                                          ⁢                                                            G                      l                                        ⁡                                          (                      t                      )                                                        ⁢                                      e                    ⁡                                          (                      t                      )                                                                                                                                              =                            ⁢                                                                                          P                      l                      T                                        ⁡                                          (                                              t                        -                        1                                            )                                                        +                                      μ                    ⁢                                                                                  ⁢                                                                  G                        l                                            ⁡                                              (                        t                        )                                                              ⁢                                                                  e                        l                                            ⁡                                              (                        t                        )                                                                              +                                      μ                    ⁢                                                                                  ⁢                                                                  G                        l                                            ⁡                                              (                        t                        )                                                              ⁢                                                                  e                        n                                            ⁡                                              (                        t                        )                                                                                            →                                                                                                      ⁢                                                P                  sl                  T                                +                                  Δ                  ⁢                                                                          ⁢                                      P                    l                    T                                                                                                          (        19        )            with parameter μ changing the speed of convergence. The linear parameters Pl are estimated with a systematic bias ΔPl if there is a correlation between the nonlinear error en(t) and the linear gradient vector Gl(t).
Minimizing the total error in the cost function in Eq. (15) may also cause a systematic bias in the estimation of the nonlinear parameters Pn. Inserting Eq. (1) and (8) into Eq. (11) and multiplying with the transposed gradient vector GnT(t) results in the Wiener-Hopf-equation for the nonlinear parameters:PnE{Gn(t)GnT(t)}=E{ynlin(t)GnT(t)}−E{[el(t)+er(t)]GnT(t)}PnSGGn=SyGn=SyGn+SrGn  (20)where SGGn is the autocorrelation of the gradient signals, SyGn is the cross-correlation between the gradients and the signal ynlin(t) and Srgn is the cross-correlation of the residual error er with the gradient signals. The nonlinear parameters of the model can directly be calculated by inverting the matrix SGGn:Pn=(SyGn+SrGn)SGGn−1Pn=Psn+ΔPn  (21)or iteratively by using the LMS-algorithm:
                                                                                          P                  n                  T                                ⁡                                  (                  t                  )                                            =                            ⁢                                                                    P                    n                    T                                    ⁡                                      (                                          t                      -                      1                                        )                                                  +                                  μ                  ⁢                                                                          ⁢                                                            G                      n                                        ⁡                                          (                      t                      )                                                        ⁢                                      e                    ⁡                                          (                      t                      )                                                                                                                                              =                            ⁢                                                                                          P                      n                      T                                        ⁡                                          (                                              t                        -                        1                                            )                                                        +                                      μ                    ⁢                                                                                  ⁢                                                                  G                        n                                            ⁡                                              (                        t                        )                                                              ⁢                                                                  e                        n                                            ⁡                                              (                        t                        )                                                                              +                                      μ                    ⁢                                                                                  ⁢                                                                                            G                          n                                                ⁡                                                  (                          t                          )                                                                    ⁡                                              [                                                                                                            e                              l                                                        ⁡                                                          (                              t                              )                                                                                +                                                                                    e                              r                                                        ⁡                                                          (                              t                              )                                                                                                      ]                                                                                            →                                                                                                      ⁢                                                P                  sn                  T                                +                                  Δ                  ⁢                                                                          ⁢                                                            P                      n                      T                                        .                                                                                                          (        22        )            
These techniques known in prior art generate a systematic deviation ΔPn from the true parameter values if either the linear error el or the residual error er correlates with the nonlinear gradient Gn:E{Gn(t)[el(t)+er(t)]}≠0  (23)
The bias ΔPn in the estimation of Pn is significant (>50%) if the nonlinear distortion ynlin is small in comparison to the residual signal er(t), which is mainly caused by imperfections in the linear modeling.
To cope with this problem, the prior art increases the complexity of the linear model (e.g. the number of taps in an FIR-filter) to describe the real impulse response hm(t) more completely. This demand can not be realized in many practical applications. For example, the suspension in a loudspeaker has a visco-elastic behavior which can hardly be modeled by a linear filter of reasonable order. The eddy currents induced in the pole plate of a loudspeaker also generate a high complexity of the electrical input impedance. In addition, loudspeakers also behave as time varying systems where aging and changing ambient conditions (temperature, humidity) cause a mismatch between reality and model which increases the residual error signal er(t).